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Fix a base field $k$. First let me loosely describe the BRST reduction in the finite-dimensional setting. For a finite-dimensional Lie algebra $\mathfrak{n}$, we can form the Clifford algebra $\mathcal{Cl}_\mathfrak{n}=\mathcal{Cl}(\mathfrak{n}\oplus\mathfrak{n}^*)$, where the bilinear form on $\mathfrak{n}\oplus\mathfrak{n}^*$ is given by the pairing between $\mathfrak{n}$ and $\mathfrak{n}^*$. Given an associative algebra $\mathcal{R}$ with a Lie algebra homomorphism $\mathfrak{n}\to\mathcal{R}$, we can make $\mathcal{Cl}\otimes\mathcal{R}$ a DG superalgebra, whose cohomology is called the BRST reduction of $\mathcal{R}$.

One wants to generalize the construction to the infinite-dimensional setting. Instead of the Clifford algebra itself, we need to pass to its Fock module (the so-called fermionic Fock space). Moreover, we need to deal with the topology of infinite-dimensional vector spaces very carefully. Beilinson et al. use the notion of Tate vector space to handle the problem.

I want to ask for references

  1. References on Tate's linear algebra (Tate vector space and Tate central extension);
  2. References on infinite-dimensional BRST reduction (better if it's written in the language of Tate's linear algebra).
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    $\begingroup$ A comprehensive treatment which might also be the latest word on the subject is Sam Raskin's Homological Algebra in Semi-Infinite Contexts, available at web.ma.utexas.edu/users/sraskin . I would also recommend the series of papers of Braunling-Groechenig-Wolfson. $\endgroup$
    – David Ben-Zvi
    Mar 2, 2023 at 17:10
  • $\begingroup$ @DavidBen-Zvi Thank you David. I also want to know if W-algebra, which is an important part of the geometric Langlands program, can be described using this Tate version BRST reduction. I understand the usual construction of W-algebra by quantized Drinfeld--Sokolov reduction, but it seems that most author doesn't use this chiral algebra way by Beilinson, Drinfeld... Instead, they write down an explicit formula without caring too much about the topology. $\endgroup$
    – Estwald
    Mar 3, 2023 at 16:22
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    $\begingroup$ Again I think you want Raskin for the most "modern" treatment of W-algebras -- W-Algebras and Whittaker Categories, available at the same link. $\endgroup$
    – David Ben-Zvi
    Mar 3, 2023 at 16:31

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